Gaussian process regression for Bayesian inverse problems

基于偏微分方程的数学模型在科学和工程中随处可见。模型中的参数，例如出现在边界条件和系数中，通常不完全已知，并且必须从观测数据估计。参数的准确重建以及重建中不确定性的估计对于可靠的预测和风险评估至关重要。一个应用实例是地下碳储存，其中环境的精确构成，如不同岩石层的位置和水力传导率，并不完全清楚，必须通过测量来估计。从数学上讲，从数据中学习未知模型参数的过程，也称为模型校准，可以用公式表示为从噪声、间接观测中恢复未知参数的逆问题。根据贝叶斯方法，我们得到了一个后验分布的未知参数的条件下观察到的数据。由于现代应用中模型的复杂性，迫切需要开发有效的算法来探索后验问题，本论文的总体目标是设计、分析和实现基于高斯过程回归的计算方法，以准确有效地求解偏微分方程反问题。我们将汇集数值分析、逼近理论、统计学和优化的想法，开发可在一系列领域应用和适应的复杂算法，包括在健康（医学成像）和环境（地下建模）领域应用的巨大潜力地下碳储存）。

Mathematical models based on partial differential equations appear everywhere in science and engineering. Parameters in the models, appearing for example in boundary conditions and coefficients, are typically not fully known and have to be estimated from observed data. Accurate reconstruction of the parameters, as well as an estimate of the uncertainty in the reconstruction, are crucial for reliable predictions and risk assessments. An example application is underground carbon storage, where the precise make-up of the environment, such as the location and hydraulic conductivity of different layers of rock, is not fully known and has to be estimated from measurements. For safety reasons, it is crucial to quantify the risk of leaked particles re-entering the human environment.Mathematically, the process of learning unknown model parameters from data, also known as model calibration, can be formulated as the inverse problem to recover unknown parameters from noisy, indirect observations. Following the Bayesian approach, we obtain a posterior distribution for the unknown parameters conditioned on the observed data. Because of the complexity of models involved in modern applications, there is a pressing need to develop efficient algorithms for exploring the posterior.The overall aim of this proposal is to design, analyse and implement computational methods based on Gaussian process regression to solve inverse problems in partial differential equations accurately and efficiently. We will bring together ideas from numerical analysis, approximation theory, statistics and optimisation, to develop sophisticated algorithms that can be applied and adapted across a range of sectors, including huge potential for applications in health (medical imaging) and environment (subsurface modelling, underground carbon storage).